Large Countable Ordinals

LCO.txt

ω^ω:
0 < 1 < 2 < 4 < 8 < ... < 3 < 6 < 12 < 24 < ... 9 < 18 < 36 < 72 < ...
  < 5 < 10 < 20 < ... < 15 < 30 < 60 < ... 45 < 90 < 180 < ...
  < 25 < 50 < 100 < ...
  
      < 7 < 14 < 28 < 56 < ...
      
          < 11 < 22 < 44 < 88 < ...
          
... and so on. 
      
The construction is to embed ω^ω into ω using prime factorization, and then compare lexicographically. We can go further.

ω^ω^ω:
0 < 1 < 2 < 4 < 16 < 256 < ... 8 < 64 < 2^12 < 2^24 < ... < 512 < 2^18 < ...
  < 32 < 2^10 < 2^20 ...
     < 128 < 2^14 < ...
     < 2^11 < 2^22 < ...
     
        < 3 < 6 < 12 < 48 < 768 < ... 
        
This is similar, but we're using the ω^ω-order above on the exponents. 

ε_0:

0 <
1 < 3 < 5 < 7 < ...                       // 2^0 * (2M + 1) in ω ordering of M
2 < 6 < 10 < 18 < 34 < 66 < ... 14 < ...  // 2^1 * (2M + 1) in ω^ω ordering of M
4 < 12 < 20 < 36 < 132 < 2052 < ...       // 2^2 * (2M + 1) in ω^ω^ω ordering of M
8 < ...                                   // 2^3 * (2M + 1) in ω^ω^ω^ω ordering of M

This is an ordering of ω + ω^ω + ω^ω^ω + ..., which appears distinct from ε_0 but is actually equivalent, analogous to how 1 + ω = ω (and NOT ω + 1).